本文深入探讨了矩阵范数的概念，包括其定义、性质及不同类型的矩阵范数，如最大绝对列和范数、谱范数和最大绝对行和范数。文章还介绍了计算矩阵-p范数的挑战，并讨论了如何通过非线性优化问题求解。
Matrix Norm
Given a square complex or real matrix 
, a matrix norm 
is a nonnegative number associated with 
having the properties
1. 
when 
and 
iff 
,
2. 
for any scalar 
,
3. 
,
4. 
.
Let 
, ..., 
be the eigenvalues of 
, then
 | (1) |
The matrix 
-norm is defined for a real number 
and a matrix 
by
 | (2) |
where 
is a vector norm. The task of computing a matrix 
-norm is difficult for 
since it is a nonlinear optimization problem with constraints.
Matrix norms are implemented as Norm[m, p], where 
may be 1, 2, Infinity, or "Frobenius".
The maximum absolute column sum norm 
is defined as
 | (3) |
The spectral norm 
, which is the square root of the maximum eigenvalue of 
(where 
is the conjugate transpose),
 | (4) |
is often referred to as "the" matrix norm.
The maximum absolute row sum norm is defined by
 | (5) |
, 
, and 
satisfy the inequality
 |
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